The change of base formula is the logarithmic identity log_b(x) = log_c(x) / log_c(b), which rewrites a logarithm in any base b as a ratio of logarithms in a base c you choose. In AP Precalculus Topic 2.13, it lets you evaluate logs your calculator can't compute directly and rewrite exponential expressions in new bases.
The change of base formula says log_b(x) = log_c(x) / log_c(b). In plain terms, if you need a logarithm in an inconvenient base b, you can swap it for a fraction of two logarithms in any base c you like, usually base 10 or base e, since those are the ones your calculator has buttons for. So log_3(50) becomes ln(50)/ln(3), and suddenly an unsolvable-looking number is just a quick calculation.
The formula has a twin that shows up in the CED for Topic 2.13. The essential knowledge for [AP Pre Calc 2.13.A] states that b^x = c^((log_c b)(x)). That identity is change of base for exponential functions. It says any exponential function can be rewritten with a different base, as long as you scale the exponent by log_c(b). Both versions are doing the same job, translating between bases so you can pick whichever base makes the problem easiest.
This lives in Unit 2: Exponential and Logarithmic Functions, specifically Topic 2.13: Exponential and Logarithmic Equations and Inequalities, supporting [AP Pre Calc 2.13.A] (solve exponential and logarithmic equations and inequalities). The CED says properties of logarithms are tools for solving these equations, and change of base is the property that bridges the gap between the base a problem hands you and the base you can actually work with. An equation like 3^x = 50 can't be solved by matching bases, so you take a log of both sides and end up with x = log(50)/log(3), which is the change of base formula in action. The exponential version, b^x = c^((log_c b)(x)), also matters because rewriting an exponential in base e or base 2 can reveal helpful information about a model, which is exactly the kind of rewriting the CED calls out.
Keep studying AP® Precalculus Unit 2
Properties of logarithms (Unit 2)
Change of base is one of the log properties, and it teams up with the product, quotient, and power rules constantly. A typical solve uses the power rule to pull an exponent down, then change of base to get a number out of the result.
Exponential equations (Unit 2)
When an exponential equation has mismatched bases, like 3^x = 50, change of base is the exit strategy. You take a log of both sides and the answer naturally lands in the form log(50)/log(3).
Properties of exponents (Unit 2)
The identity b^x = c^((log_c b)(x)) only works because of exponent rules, since (c^(log_c b))^x = c^((log_c b)x). Change of base for exponentials is really an exponent property and a log property shaking hands.
Extraneous solutions (Unit 2)
After you change bases and solve, the CED still expects you to check your answer against domain restrictions. Logarithms only accept positive inputs, so a solution that makes any log's argument zero or negative gets thrown out.
Multiple-choice questions hit this two ways. First, the classic solve. A question gives you something like 3^x = 50, points out that you can't match the bases, and asks which method works. The answer is to take a logarithm of both sides, which produces x = log(50)/log(3). Second, the identity version. Questions test whether you recognize that c^(log_c b) = b, asking things like "if e^(kx) = (2^(log_2 e))^(kx), what is log_2 e?" These look intimidating but collapse fast once you see that 2^(log_2 e) is just e. No released FRQ has used the phrase "change of base formula" verbatim, but Part A (calculator-allowed) FRQs routinely involve solving exponential equations where your calculator answer comes from exactly this formula. Know it both directions, log form and exponential form.
Change of base says log_b(x) = log_c(x)/log_c(b), a ratio of two separate logs. The quotient rule says log_c(x/b) = log_c(x) - log_c(b), one log of a fraction turning into a difference. Mixing them up is a classic trap. log(50)/log(3) is NOT log(50/3) and it's NOT log(50) - log(3). Division of logs and logs of division are completely different things.
The change of base formula is log_b(x) = log_c(x)/log_c(b), and the new base c can be anything, though base 10 or base e is the practical choice for calculators.
Its exponential twin, b^x = c^((log_c b)(x)), appears in the CED for Topic 2.13 and lets you rewrite any exponential function in a new base.
The shortcut identity c^(log_c b) = b is the engine behind change of base, and AP multiple-choice questions test whether you spot it inside scary-looking expressions.
An equation like 3^x = 50 can't be solved by matching bases, so you take a log of both sides and the solution comes out as log(50)/log(3).
Dividing two logs is not the same as the log of a quotient. log(50)/log(3) does not equal log(50/3) or log(50) - log(3).
After changing bases and solving, check for extraneous solutions, since log arguments must stay positive.
It's the identity log_b(x) = log_c(x)/log_c(b), which rewrites a logarithm in base b as a ratio of logarithms in a new base c. It supports learning objective 2.13.A in Unit 2, solving exponential and logarithmic equations.
No. log(50)/log(3) is the change of base formula and equals log_3(50), roughly 3.56. log(50/3) uses the quotient rule and is a totally different number, about 1.22. Division of logs and logs of division are not interchangeable.
Yes. The exam doesn't give you a formula sheet of log identities, and the CED for Topic 2.13 explicitly includes the related identity b^x = c^((log_c b)(x)). You should be able to use both the log version and the exponential version.
Change of base turns one log into a fraction of two logs in a new base, log_b(x) = log_c(x)/log_c(b). The quotient rule turns the log of a fraction into a difference, log_c(x/b) = log_c(x) - log_c(b). One outputs division, the other outputs subtraction, and they answer different questions.
You can't write both sides with the same base, so take a logarithm of both sides. That gives x·log(3) = log(50), so x = log(50)/log(3) ≈ 3.56. The result is literally log_3(50) rewritten by change of base.
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